Electric Flux Calculator

Electric flux measures how much electric field passes through a surface:

Φ = E × A × cos(θ)

And Gauss’s law relates flux to enclosed charge:

Φ = Q_enclosed / ε₀

Where:

• Φ = Electric flux (N·m²/C or V·m)
• E = Electric field strength (N/C = V/m)
• A = Area of the surface (m²)
• θ = Angle between the electric field vector and the surface normal
• Q_enclosed = Total charge enclosed by the Gaussian surface (C)
• ε₀ = Permittivity of free space = 8.854 × 10⁻¹² C²/(N·m²)

Understanding the angle θ:

• θ = 0°: Field is perpendicular to the surface (maximum flux)
• θ = 90°: Field is parallel to the surface (zero flux — field “slides along” the surface)
• θ = 180°: Field is anti-parallel to surface normal (negative flux)

Gauss’s Law application: The total electric flux through any closed surface equals the enclosed charge divided by ε₀. This makes calculating electric fields from symmetric charge distributions very easy:

• Sphere of charge: flux = Q/ε₀ regardless of sphere size
• Infinite plane of charge: flux through box = σA/ε₀

Physical intuition: Think of electric flux as counting how many electric field lines pass through a surface. More field lines means either a stronger field or a larger area, and either way more flux.

A worked example

Hold a 0.5 m² sheet square-on to a 1,000 N/C field. Φ = 1000 × 0.5 × cos(0°) = 500 N·m²/C.

Now tilt it to 60°. cos(60°) = 0.5, so the flux halves to 250 N·m²/C, even though nothing about the field or the sheet has changed. Tilt to 90° and it drops to zero: the field runs along the surface without ever crossing it. This is why the angle is not a detail you can skip.

The open-surface trap

The calculator’s Gauss line deserves a warning. Gauss’s law says the flux through a closed surface equals the enclosed charge over ε₀, and the word closed is doing all the work. A flat sheet is open, and field lines that enter it can leave again round the edges without being counted.

So if you compute 500 N·m²/C through one square patch and multiply by ε₀, you do not get “the charge behind the patch”. You get a number with no physical meaning. The Gauss step is only valid when the flux you started with was summed over a surface that completely surrounds the charge: a sphere, a cube, a cylinder with both end caps.

Why Gauss’s law is worth the trouble

For a symmetric charge distribution it turns a hard integral into arithmetic. A sphere of total charge Q gives flux Q/ε₀ through any concentric sphere, whatever its radius, so E must fall as 1/r² outside the charge. That single argument reproduces Coulomb’s law without integrating anything.

It also gives one of the most useful results in electrostatics: the field inside a hollow conductor is zero. Draw a Gaussian surface inside the metal, note that no charge is enclosed, and the flux must vanish. That is why a car protects you from lightning and why sensitive electronics sit inside metal cans.